From Gogeometry

Using :

• Isometric transformation
• Inscribed angles in a circle
• Equilateral triangle
• Tangents to a circle

• Define R radius of circle O and r radius of circle Q
• By construction and symmetry
• A, O, Q, D are collinear
• DE=DF, AE=AF
• AD angle bisector of ∠BAC
• D symmetric of A by O => ED=EA=FD=FA : AEDF is a diamond
  =>O middle of EF and EF ⊥ AD
• ΔABC equilateral and AD angle bisector of BAC => ∠CAD=Π/6
• ∠FAO=Π/6 and ∠AOF=Π/2 => ∠AFO=Π/3
• ∠AFO intercepts arc FE of circle Q => ∠EDF=Π/3
• ∠EDF=Π/3 and DE=DF => ΔEDF is equilateral
• QD=r, OD=R, ΔEDF is equilateral
  => QD=2OD/3 => r=2R/3 and R/r=3/2
• S=[Circle O]-[Circle Q], S1=[Circle Q]=> S/S1=[Circle O]/[Circle Q]-1
  => S/S1=(R/r)^2-1=9/4-1
  Therefore S/S1=5/4